J. of Mult.-Valued Logic & Soft Computing., Vol. 11, pp. 567–602 Reprints available directly from the publisher Photocopying permitted by license only  c 2005 Old City Publishing, Inc. Published by license under the OCP Science imprint, a member of the Old City Publishing Group Ternary GFSOP Minimization using Kronecker Decision Diagrams and Their Synthesis with Quantum Cascades Mozammel H. A. Khan ∗,† , Marek A. Perkowski ∗∗ , Mujibur R. Khan ∗ and Pawel Kerntopf ‡ ∗ Department of Computer Science and Engineering, East West University, 43 Mohakhali, Dhaka 1212, BANGLADESH, Email: mhakhan@ewubd.edu ∗∗ Department of Electrical and Computer Engineering, Portland State University, 1900 SW 4 th Avenue, Portland, OR 97201, USA ‡ Institute of Computer Science, Warsaw University of Technology, Nowowiejska 15/19, 00-665 Warsaw, POLAND Ternary Galois Field Sum of Products (TGFSOP) expressions are found to be a good choice for ternary reversible logic and particularly for quantum cascaded realization of ternary functions. In this paper, we propose 5 ternary shift operations and various basic and composite ternary literals for defining TGFSOP expression. We propose 16 Ternary Galois Field Expansions (TGFE) using these literals and three new types of Ternary Galois Field Decision Diagrams (TGFDD) using the proposed expansions, which are useful for reversible and quantum logic design. We also propose a heuristic for creating optimal Kronecker TGFDD and methods for flattening the TGFDDs for determining near-minimum TGFSOP expressions. Besides, we propose quantum realizations for the 5 ternary Shift gates and a ternary swap gate. We also propose a new generalization of ternary Toffoli gates with their implementation from truly realizable 2-qudit quantum primitives. Further, we propose a method of synthesizing multi-output TGFSOP using cascade of ternary Shift gates, Swap gate, and generalized Toffoli gate. Finally, we present experimental results to show the complexity of the decision diagrams, the resultant TGFSOP expressions, and the new quantum cascade for some ternary benchmark functions. † Corresponding author. 567